Immutability is not bad, there are some situations you want immutability. For example, to secure voting systems, you may want to be able to write on the chain that "precinct 156 reported votes x/y/z in this quantity" so that if anybody comes along and tampers with those numbers later on, you can point to the chain and say "no see, actually, these are the real original numbers that the precinct published". The precinct could lie about their numbers of course and publish bad numbers to the chain, blockchain doesn't protect against that (unless the votes themselves are recorded on the chain by the individual voter), but the blockchain protects against those numbers changing in the future or another party incorrectly claiming they are a/b/c when they are actually x/y/z. That's a situation where immutability helps. Same with financial transactions. If you sent somebody money, you want a record of that (a receipt) if they later claim you never sent it to them. Examples of records which have a high degree of immutability that people use in everyday life are: court records, census data, house deeds, etc.
Blockchains usually have some degree of immutability but from a technical perspective they don't necessarily have to. If we're talking about data storage, you don't have to store the data itself on the chain, the chain data can just "point to" off-chain data which you can take down or modify at will.
An example of a scenario where this could work is: you have a blockchain for coordinating the sharing of medical information between different parties. You, as a user, have an account on this blockchain. The only data stored on chain is a list of parties and who you have authorized to receive your medical data along with a pointer to a file storage system like Amazon AWS which contains your medical data in encrypted format. You can add or revoke authorization at any time by changing how that data is encrypted. Whoever you gave authorization to prior may have made a copy of the data at that point in time, but you can block them from accessing new data you put out. When Amazon AWS gets a request to transfer a copy of your data to a new party, they can check the blockchain to see if that party is authorized to receive it.
The benefit of such a system would be that:
Your medical records are yours and stored in your own data storage system over which you have complete control.
You can choose to share it with parties like insurance providers or researchers who need large medical data sets to comb through.
You could set this control at a very granular level or grant access to all your data.
Your data becomes portable between insurance providers and your insurance provider can't make your life difficult by refusing to export data to your new one.